18 . 024 – ESG Notes 2 Pramod

Inverses and determinants of matrices are closely related and come up in many different contexts. In these notes we will develop some basics of the theory of these ideas. Our initial motivation is the solution of systems of linear equations, such as 2x − 5y + 4z = −3 x − 2y + z = 5 x − 4y + 6z = 10. To solve the system, we want to manipulate these equations to get new equations which only contain one variable each. What do we mean by “manipulate”? We can add (multiples of) one equation to another, or we can multiply both sides of a single equation by a constant. And, of course, we are allowed to rearrange the order of the equations if we want to. This process is encoded into a shorthand notation by the method of Gauss-Jordan elimination. We write down just the coefficients of the above system in an augmented matrix,